The difference between two numbers is 3 and the difference between their cubes is 999. Find the difference between their squares.

Options

A. 63
B. 36
C. 81
D. 18

Answer

To solve the problem, we need to find the difference between the squares of two numbers given the following conditions:

1. The difference between the two numbers is 3.
2. The difference between their cubes is 999.

Let's denote the two numbers as \( X \) and \( Y \).

### Step-by-Step Solution:

**Step 1: Set up the equations based on the given information.**

From the problem, we have:
- \( X - Y = 3 \) (Equation 1)
- \( X^3 - Y^3 = 999 \) (Equation 2)

**Step 2: Use the identity for the difference of cubes.**

We can use the identity for the difference of cubes:
\[ X^3 - Y^3 = (X - Y)(X^2 + XY + Y^2) \]

Substituting Equation 1 into this identity gives us:
\[ 999 = (3)(X^2 + XY + Y^2) \]

**Step 3: Simplify to find \( X^2 + XY + Y^2 \).**

Dividing both sides by 3:
\[ X^2 + XY + Y^2 = \frac{999}{3} = 333 \quad (Equation 3) \]

**Step 4: Express \( X^2 + Y^2 \) in terms of \( XY \).**

We know that:
\[ X^2 + Y^2 = (X - Y)^2 + 2XY \]
Substituting \( X - Y = 3 \):
\[ X^2 + Y^2 = 3^2 + 2XY = 9 + 2XY \]

**Step 5: Substitute into Equation 3.**

Now, substituting \( X^2 + Y^2 \) into Equation 3:
\[ 9 + 2XY + XY = 333 \]
This simplifies to:
\[ 9 + 3XY = 333 \]

**Step 6: Solve for \( XY \).**

Subtracting 9 from both sides:
\[ 3XY = 324 \]
Dividing by 3:
\[ XY = 108 \quad (Equation 4) \]

**Step 7: Find the difference between the squares.**

We need to find \( X^2 - Y^2 \). We can use the identity:
\[ X^2 - Y^2 = (X + Y)(X - Y) \]

**Step 8: Find \( X + Y \).**

We can find \( X + Y \) using the equations we have. From Equation 1:
\[ X = Y + 3 \]
Substituting into the expression for \( XY \):
\[ (Y + 3)Y = 108 \]
This expands to:
\[ Y^2 + 3Y - 108 = 0 \]

**Step 9: Solve the quadratic equation.**

Using the quadratic formula \( Y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ Y = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 1 \cdot (-108)}}{2 \cdot 1} \]
\[ Y = \frac{-3 \pm \sqrt{9 + 432}}{2} \]
\[ Y = \frac{-3 \pm \sqrt{441}}{2} \]
\[ Y = \frac{-3 \pm 21}{2} \]

Calculating the two possible values for \( Y \):
1. \( Y = \frac{18}{2} = 9 \)
2. \( Y = \frac{-24}{2} = -12 \) (not applicable since we are looking for positive numbers)

Thus, \( Y = 9 \) and substituting back gives \( X = 12 \).

**Step 10: Calculate \( X + Y \) and \( X - Y \).**

Now we can find:
\[ X + Y = 12 + 9 = 21 \]
\[ X - Y = 12 - 9 = 3 \]

**Step 11: Calculate \( X^2 - Y^2 \).**

Now substituting into the difference of squares:
\[ X^2 - Y^2 = (X + Y)(X - Y) = 21 \cdot 3 = 63 \]

### Final Answer:
The difference between their squares is \( 63 \).